Semantics & Pragmatics Volume 5, Article 1: 1–25, 2012
http://dx.doi.org/10.3765/sp.5.1
Two types of class B numeral modifiers:
A reply to Nouwen 2010∗
Bernhard Schwarz
McGill University
Brian Buccola
McGill University
Michael Hamilton
McGill University
Received 2011-07-29 / First Decision 2011-09-12 / Revisions Received 2011-11-14 /
Second Decision 2011-11-21 / Published 2012-02-13
Abstract Nouwen (2010) proposes that numeral modifiers like at most and
up to belong to a natural class of expressions that share the same semantic
interpretation. We identify several semantic contrasts between at most and
up to that are inconsistent with this proposal. Moreover, we show that
Nouwen’s semantics cannot be right for at most. In attempting to apply
his semantics to up to, we propose an amendment intended to accomodate
certain observations not captured in Nouwen’s original account.
Keywords: modified numerals, degree scales, ignorance implications, monotonicity,
weak numerals
∗ For discussion of material related to this article, we thank the participants of three syntaxsemantics research group meetings at McGill during the summer and fall of 2011. We also
thank two anonymous referees, as well as David Beaver, for comments that have helped us
improve the final version. The first author was supported in part by SSHRC Standard Research
Grant 410-2010-1264 (principal investigator: Junko Shimoyama) and FQRSC Programme
Soutien aux Équipes de Recherche grant 2012-SE-144646 (principal investigator: Lisa Travis).
The second author received support from SSHRC Standard Research Grant 410-2010-1254
(principal investigator: Brendan Gillon), and a McCall McBain Fellowship from the McGill
Faculty of Arts. The third author received support from SSHRC Research Grants 410-20111062 (principal investigator: Michael Wagner) and 410-2008-1687 (principal investigator: Lisa
Travis), as well as SSHRC Strategic Grant 869-2009-0004 (principal investigator: Michael
Wagner).
©2012 Bernhard Schwarz, Brian Buccola, and Michael Hamilton
This is an open-access article distributed under the terms of a Creative Commons NonCommercial License (creativecommons.org/licenses/by-nc/3.0).
Bernhard Schwarz, Brian Buccola, and Michael Hamilton
1 Introduction
So-called numeral modifiers combine with numerals into expressions like
less than ten, at most five, etc. The focus of Nouwen 2010 is an interesting
contrast between two types of numeral modifiers, illustrated in (1) and (2).
(1)
A hexagon has fewer than 11 sides.
(2)
a. #A hexagon has at most 10 sides.
b. #A hexagon has up to 10 sides.
A
B
B
(Nouwen 2010: 3:3)
Nouwen attributes the acceptability contrast between the sentences in (1)
and (2) to a semantic contrast between the numeral modifiers they contain.
Specifically, he takes the numeral modifiers in (2) to introduce an implication
of speaker ignorance as to the number of sides a hexagon has. Nouwen
attributes the oddness of the particular examples in (2) to the implausibility
of this implication, which (1) lacks.1
Nouwen refers to numeral modifiers that introduce ignorance implications
as class B modifiers and those that do not as class A modifiers. Furthermore,
Nouwen distinguishes upper-bound numeral modifiers, such as those in (1)
and (2), from lower-bound numeral modifiers, such as more than, at least and
minimally.
In this reply we focus on Nouwen’s analysis of upper-bound class B
numeral modifiers. Nouwen argues that all upper-bound class B numeral
modifiers share the same interpretation. We present data that challenge this
proposal, identifying two types of semantic contrasts between at most and up
to: one involving NPI licensing and the corresponding inferences and another
involving what we refer to as the bottom-of-the-scale effect. We explore how
Nouwen’s analysis applies to these two upper-bound class B modifiers. We
argue that it fails for at most, but we explore how a conservative amendment
of his analysis, extended with what we call the range requirement, applies to
up to. One consequence, if our amendment is on the right track, is that the
ignorance implications for at most and up to have different sources.
In section 2, we review Nouwen’s analysis of upper-bound class B numeral
modifiers. Section 3 identifies a contrast between at most and up to with
regard to NPI licensing and the corresponding inferences. Section 4 describes
1 We take Nouwen’s use of “#” in (2) to indicate mere implausibility, as opposed to downright
unacceptability. In later sections, we apply “#” differently to mark semantic/pragmatic
ill-formedness in our own examples.
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Two types of class B numeral modifiers
another contrast with regard to the bottom-of-the-scale effect, and we provide
a characterization of it within the framework employed by Nouwen. In section
5 we propose and explore an amended semantics for up to which incorporates
the range requirement. We show that, apart from capturing the bottomof-the-scale effect, it has the welcome consequence of deriving ignorance
implications. In section 6 we identify and discuss a point of tension between
our account and the existence of weak interpretations of numerals. Section 7
concludes.
2
Nouwen’s semantics for class B numeral modifiers
2.1 Bare numerals
Following Hackl 2000, Nouwen (2010) proposes that bare numerals, such as
three in (3), serve as the argument of a silent counting quantifier, many, as
shown in the logical form or LF in (4).
(3)
Three students attended.
(4)
[[three many] students] attended
Nouwen assigns to many the denotation in (5), where X is a singular or plural
individual and |X| is the number of singular parts of X.2
(5)
‚manyƒ = λdd .λfet .λget .∃!X[ |X| = d & f(X) = g(X) = T ]
Under this semantics for many, the LF in (4) will be assigned the truth
conditions in (6), which states that exactly three students attended.
(6)
∃!X[ |X| = 3 & ‚studentsƒ(X) = ‚attendedƒ(X) = T ]
2.2 Upper-bound class B numeral modifiers and covert modality
Following Hackl 2000, Nouwen (2010) assumes that a numeral modifier
combines with a numeral to form a degree phrase denoting a generalized
quantifier over degrees. For interpretability, this degree phrase must raise
2 Actually, Nouwen introduces two numeral counting quantifiers: a strong many (which he
calls many2 ), with the denotation in (5), and a weak many (many1 ), as in Hackl 2000. Since the
weak many is not central to Nouwen’s account of numeral modifiers, nor to our discussion
of it, we disregard it until section 6.
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Bernhard Schwarz, Brian Buccola, and Michael Hamilton
covertly to derive a degree predicate in its scope. Under these assumptions,
(8) is an interpretable LF for the example in (7).
(7)
At most three students attended.
(8)
[[at most] three] λd[[[d many] students] attended]
The lexical entry that Nouwen proposes for upper-bound class B numeral
modifiers like at most and up to is shown in (9), according to which these
modifiers equate the degree denoted by the numeral with the maximum
element of the set given by the degree predicate.
(9)
‚at most/up toƒ = λdd .λfdt .max(f) = d
Under this semantics for upper-bound class B numeral modifiers, the LF
in (8) is assigned the truth conditions in (10).
(10)
max({d : ∃!X[ |X| = d & ‚studentsƒ(X) = ‚attendedƒ(X) = T ]}) = 3
The set of degrees to which max applies in (10) is the singleton set containing
the cardinality of the set of students who attended, i.e., the exact number of
attending students.3 Of course, the maximum element of a singleton set is its
single element. Accordingly, (10), like (6), states that exactly three students
attended.
To be sure, this is not an attested interpretation of (7), nor does Nouwen
claim that it is. Nouwen credits the actual truth conditions of (7) to a different
LF, namely the one in (11). This LF differs from (8) only in that “♦”, a silent
speaker-oriented epistemic possibility operator, appears in the immediate
scope of the degree phrase.
(11)
[[at most] three] λd[♦ [[[d many] students] attended]]
The LF in (11) receives the truth conditions in (12), which states that the
maximal number of students who could have attended, according to the
speaker’s evidence, is three. Nouwen suggests that this is the intended
interpretation of (7).
(12)
max({d : ♦∃!X[ |X| = d & ‚studentsƒ(X) = ‚attendedƒ(X) = T ]}) = 3
3 We are putting aside, for now, the problematic case where the set of students who attended
is empty. We return to this issue in section 4, where it will be important.
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Two types of class B numeral modifiers
Two questions remain. First, what excludes the unattested truth conditions in (10), hence what excludes the LF in (8)? Nouwen proposes that the
exclusion of this LF is a blocking effect of the sort familiar from the morphophonological literature (e.g., Kiparsky 1973, 1983). Specifically, Nouwen takes
the LF in (8) to be blocked by the truth conditionally equivalent but less
marked LF in (4).4
Second, how does Nouwen’s account derive the ignorance implications
which he takes to be a defining feature of class B numeral modifiers? Surprisingly, this is not addressed. Note that, (12), as it stands, does not entail
speaker ignorance regarding the number of students who attended. We
return to this issue in section 5.
3
NPI licensing and downward entailment
3.1
At most
It has been observed in the literature that phrases headed by at most license
negative polarity items (NPIs), such as ever in (13) and give a damn in (14).
(13)
At most three people had ever been in this cave.
(Krifka 2007)
(14)
At most three students give a damn about Pavarotti.
(Chierchia & McConnell-Ginet 2000: 522)
This observation puts Nouwen’s (2010) analysis of upper-bound class B
modifiers in conflict with Ladusaw’s (1979) classic account of NPI licensing,
which equates the class of NPI licensors with the class of downward entailing
expressions. Under Nouwen’s account, at most three is clearly not downward
entailing; in fact, it is non-monotone. Obviously, for any two sets of degrees
A and B such that A is a subset of B, nothing guarantees that max(A) equals
max(B).
Moreover, it would not be possible for Nouwen to resolve this conflict
by simply rejecting Ladusaw’s analysis of NPI licensing. Putting aside NPIs,
intuitions on inferences alone suggest that at most three should be analyzed
as downward entailing. The validity of relevant inferences, like the one from
(15a) to (15b), seems to be uncontroversial.5
4 Even though Nouwen doesn’t present a general theory of markedness, it indeed seems
plausible to us to assume that (8) is more marked than (4).
5 Actually, in view of the ignorance implications with class B modifiers described by Nouwen
(see section 1 above), the intuitive validity of this inference is not necessarily expected. To
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Bernhard Schwarz, Brian Buccola, and Michael Hamilton
(15)
a.
b.
At most three students smoke.
At most three students smoke cigars.
(Chierchia & McConnell-Ginet 2000: 522)
And yet, by virtue of the non-monotone semantics Nouwen assigns to at most
three, the satisfaction of the truth conditions of (16a), shown in (17a), does
not guarantee the satisfaction of the truth conditions of (16b), shown in (17b).
(16)
a.
b.
[[at most] three] λd[♦ [[[d many] students] smoke]]
[[at most] three] λd[♦ [[[d many] students] [smoke cigars]]]
(17)
a.
b.
max({d : ♦∃!X[ |X| = d & ‚studentsƒ(X) = ‚smokeƒ(X) = T ]}) = 3
max({d : ♦∃!X[ |X| = d & ‚studentsƒ(X) = ‚smoke cigarsƒ(X) = T ]})
=3
For example, in a scenario where the speaker is sure that exactly one, two
or three students smoke, while also being sure that exactly one or two (but
not three) students smoke cigars, (17a) is true, while (17b) is false. Thus, it
is clear that Nouwen cannot account for the attested inference from (15a) to
(15b).
3.2
Up to
Recall now that one of Nouwen’s (2010) main points is that upper-bound
class B modifiers, including up to and at most, share the same semantic
interpretation. It turns out that up to is markedly different from at most
when it comes to NPI licensing. (18) and (19) are unacceptable, unlike their
counterparts with at most in (13) and (14).
(18)
*Up to three people had ever been in this cave.
(19)
*Up to three students give a damn about Pavarotti.
Furthermore, in accordance with Ladusaw’s (1979) analysis of NPI licensing, up to fails to support the corresponding inferences. That is, our intuitions
indicate that (20b) cannot be inferred from (20a).
be sure, it is possible to be uncertain about the exact number of students who smoke while
being certain about the exact number of students who smoke cigars. Apparently, ignorance
implications with at most are not obligatory. Looking ahead, note that this is consistent with
our working hypothesis in section 5 that ignorance implications with up to are part of the
asserted, truth-conditional content.
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Two types of class B numeral modifiers
(20)
a.
b.
Up to three students smoke.
Up to three students smoke cigars.
Specifically, the same scenario described above readily comes to mind according to our intuitions: In a scenario where the speaker is sure that exactly
one, two or three students smoke, while also being sure that exactly one
or two (but not three) students smoke cigars, (20a) is true and appropriate,
while (20b) is not.6
These observations cast doubt on Nouwen’s proposal that all class B
modifiers share the same semantic interpretation.7 However, while we have
excluded the lexical entry in (9) for at most, it might be just what is needed
for up to. In particular, not only does Nouwen’s analysis predict the unacceptability of (18) and (19) and the invalidity of the inference from (20a) to
(20b), it also fits with our intuitions, noted above, as to why this inference is
invalid.8
6 It is clear that it would be inappropriate for a speaker to utter (20b) in the scenario provided.
However, it is not so clear that (20b) is judged to be false, even though Nouwen’s analysis
predicts that it is false. Under Nouwen’s analysis, there is a missing link between falsity
and the judgment of inappropriateness. This issue could be attributed to the fact that the
sentences in question are statements about the speaker’s epistemic state. The question,
which we leave to future research, is how predicted truth values in such cases translate into
judgments on appropriateness. Thanks to David Beaver for raising this issue.
Note also that, as predicted by the non-monotone semantics for up to, we do not judge
the reverse inference from (20b) to (20a) to be valid, either. (20a) conveys information that
(20b) does not, viz. that (the speaker is certain that) no more than three students smoke.
7 Citing Gajewski’s (2008) analysis of only and no one but, a reviewer suggests that the NPI
licensing contrast presented above could be consistent with semantic equivalence of at
most and up to if it could be argued that the two differ in internal syntactic structure. The
reviewer’s point is well taken. However, we are not sure how to spell out this suggestion, since
it appears to us that a structural contrast in the spirit of Gajewski 2008 would necessitate a
semantic contrast as well. Moreover, the following section presents an additional contrast
between up to and at most which we interpret as further indication that the two are not
semantically equivalent.
8 As illustrated by the contrast in pragmatic well-formedness between (i) and (ii) below, there
are dramatic differences in use between up to and at most.
(i)
Save up to 50% on airtime rates!
(ii)
#Save at most 50% on airtime rates!
While we leave the detailed analysis of such contrasts as a topic for future work, we speculate
that they relate to the monotonicity contrast described in this section.
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Bernhard Schwarz, Brian Buccola, and Michael Hamilton
4 Bottom-of-the-scale effect
4.1 Another contrast
The pattern in (21) and (22) illustrates another systematic contrast between at
most and up to, showing that up to is sensitive to the choice of the numeral
being modified in a way that at most is not.
(21)
(22)
a.
b.
At most ten people died in the crash.
Up to ten people died in the crash.
(Nouwen 2008: 580)
a. At most one person died in the crash.
b. #Up to one person died in the crash.
While at most combines freely with either one or a higher numeral like ten,
the use of up to in (22b) is unacceptable. A natural speculation as to the
relevant difference between one and ten is that one in some sense denotes
the bottom of the scale. The acceptability contrast in (22) suggests, then, that
up to cannot combine with a bottom-of-the-scale numeral. We refer to this
restriction on up to as the bottom-of-the-scale effect.
The observation that this effect is only attested with one of the two
modifiers under consideration further questions Nouwen’s (2010) proposal
that class B modifiers share the same semantic interpretation. In the previous
section, we concluded that Nouwen’s semantics does not apply correctly to
at most, but is still an option for up to. Therefore, for the remainder of this
paper we confine our attention to the latter.
It turns out that, as it stands, Nouwen’s analysis of class B modifiers fails
to predict the bottom-of-the-scale effect with up to. The analysis assigns
sentence (22b) the LF in (23), which receives the truth conditions in (24).
(23)
[[up to] one] λd[♦ [[d many] people died in the crash]]
(24)
max({d : ♦∃!X[ |X| = d & ‚peopleƒ(X) = ‚died in the crashƒ(X) = T ]}) = 1
The statement in (24) amounts to saying that, according to the speaker’s
evidence, no more than one person could have died in the crash. Since this
would seem to be a useful meaning for a sentence to convey, no explanation
for the unacceptability of (22b) emerges.
To accomodate the bottom-of-the-scale effect, we now present and explore
a conservative revision of Nouwen’s account.
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Two types of class B numeral modifiers
4.2
The bottom of the scale
Naturally, to explain the bottom-of-the-scale effect, we will first have to
elucidate the notion of bottom of the scale. We begin by taking a closer look
at the set of degrees in (25), the set to which max applies in (24) above.
(25)
{d : ♦∃!X[ |X| = d & ‚personƒ(X) = ‚died in the crashƒ(X) = T ]}
The smallest possible element of (25) depends in part on what exactly the
variable X is taken to range over. Nouwen’s (2010) assumptions in this
regard are unclear to us. But the issue is addressed in Hackl 2000, which
established the framework for Nouwen’s analysis of numeral modifiers. Hackl
is explicit in not assuming the existence of a zero individual, i.e., an individual
corresponding to the empty set in set theory.9 The absence of such a possible
value for X guarantees that (25) cannot have 0 as an element.
In fact, we can be certain that (25) does not contain any degree smaller
than 1: While it is perhaps possible for half a person to be photographed or
to be covered with sand, it is impossible for half a person to die in a crash.
The smallest possible degree contained in the set (25) is accordingly 1, i.e.,
the degree that we have characterized as the bottom of the scale.
The bottom of the scale, then, is the smallest possible element of the set
of degrees to which the degree phrase headed by up to applies under (our
construal of) Nouwen’s analysis. In more theory-neutral terms, a bottom-ofthe-scale numeral denotes the lowest non-zero degree that can satisfy the
degree property provided by the sentence that contains it. So 1 is the bottom
of the scale in (22b) because a fatal crash necessarily implies the death of at
least one person.
This general characterization of the bottom of the scale suggests that it
need not always be given by one, hence that the bottom-of-the-scale effect can
be reproduced with other numerals. This is indeed so. Suppose, for example,
that eggs can be bought in half-dozen cartons only. Under this assumption,
our judgments on the examples (26b) and (27b) are parallel to those we gave
for (21b) and (22b) above. In particular, we detect the same unacceptability in
(27b) as in (22b).10
9 Hackl (2000: 169) writes: “Note that the definition does not allow us to count zero individuals,
i.e., there is no individual that is assigned degree zero in a complete join semi-lattice. I will
have to leave it for future research to extend the definition so that expressions like ‘more
than zero students’, ‘fewer than one students’, etc. can be covered.”
10 Replacing a dozen and half a dozen with twelve and six, respectively, does not significantly
alter our judgments.
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Bernhard Schwarz, Brian Buccola, and Michael Hamilton
(26)
(Assumption: Eggs can be bought in half-dozen cartons only.)
a.
b.
(27)
He bought at most a dozen eggs.
He bought up to a dozen eggs.
(Assumption: Eggs can be bought in half-dozen cartons only.)
a. He bought at most half a dozen eggs.
b. #He bought up to half a dozen eggs.
Under Nouwen’s analysis, (27b) has the LF in (28), which receives the truth
conditions in (29); (30) isolates the set of degrees to which max applies in
(29).
(28)
[[up to] half a dozen] λd[♦ [he bought [d many] eggs]]
(29)
max({d : ♦∃!X[ |X| = d & ‚eggsƒ(X) = ‚he boughtƒ(X) = T ]}) = 6
(30)
{d : ♦∃!X[ |X| = d & ‚eggsƒ(X) = ‚he boughtƒ(X) = T ]}
By assumption, any purchase of eggs has to be a purchase of at least six.
Accordingly, in the absence of a zero individual, 6 is the smallest possible
element of the set of degrees in (30). Thus, half a dozen in (27b) denotes
the bottom of the scale, and so we are not surprised that (27b) shares the
oddness of (22b).
It is also possible, of course, for some scale not to have a bottom at all. In
such a case, the bottom-of-the-scale effect is not expected to arise with one
or with any degree-denoting phrase, for that matter. For example, since the
amount of cake consumed in a cake-eating event can be arbitrarily small, and
in particular less than one entire cake, one in (31) need not denote the bottom
of the scale. Hence, we correctly expect this example to be acceptable.11
(31)
She ate up to one whole cake.
11 Example (31) sounds better with whole (or entire) than without, and we speculate that this
is because whole signals the possibility of eating less than one cake, i.e., it signals that the
relevant scale reaches below 1. Note that this speculation is supported by the oddness of
the sentence One entire person died in the crash, whose content is inconsistent with a scale
reaching below 1.
Note also that, while it seems clear that the contrast between (31) and (22b) is to be
explained along the lines suggested here, the framework that Nouwen adopts from Hackl
2000 is not actually set up to support formalization of this account. The reason is that Hackl
does not countenance quantification over proper parts of a singular individual such as a
cake. For relevant discussion, including analyses of entire and whole, see Morzycki 2002 and
Moltmann 2005.
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Two types of class B numeral modifiers
5 The range requirement
Providing a general characterization of the bottom of the scale is of course
just one step towards understanding the bottom-of-the-scale effect. In this
section, we take the next step by exploring a possible derivation of this effect.
We amend Nouwen’s (2010) semantics for up to by adding an entailment
that we refer to as the range requirement. We then show that this addition
has other welcome consequences. In particular, we show that it derives the
ignorance implications described in section 1.12
5.1 Deriving the bottom-of-the-scale effect
Having identified 1 and 6 as the smallest possible elements of the sets in
(25) and (30), respectively, let us revisit the truth conditions in (24) and (29).
According to these truth conditions, the smallest possible degree in a set of
degrees, the so-called bottom of the scale, is also the maximal element of the
set. Note that this is tantamount to saying that the bottom of the scale is
the only element of the set in question. This means that (24) and (29), in our
scenario, can be rewritten as (32) and (33), respectively.
(32)
{d : ♦∃!X[ |X| = d & ‚peopleƒ(X) = ‚died in the crashƒ(X) = T ]} = {1}
(33)
{d : ♦∃!X[ |X| = d & ‚eggsƒ(X) = ‚he boughtƒ(X) = T ]} = {6}
More generally, when up to modifies a bottom-of-the-scale numeral, the
resulting truth conditions entail that the degree denoted by that numeral is
the only element of the set of degrees that the modified numeral composes
with.
12 While preparing the final version of this reply, we were reminded of a closely related proposal
by Penka (2010). Penka applies a general semantics for superlatives to at least and at most in
an attempt to compositionally derive their truth conditional contributions. By decomposing
at least and at most, Penka effectively adds a range requirement (as a presupposition) to
their semantics. Moreover, Penka intends for the range requirement to have the same
effects regarding blocking and the derivation of ignorance implications that we discuss in
sections 5.2 and 5.3 below, and analogous to our discussion in sections 6.3 and 6.4, Penka
identifies the potential problems for this approach coming from weak interpretations of
numerals (and other measure phrases). Naturally, however, being concerned with superlative
modifiers, rather than up to, Penka does not establish a link between the range requirement
and our bottom-of-the-scale effect. In fact, under Penka’s proposal, we wonder how one
might accomodate the observation, reported in section 4, that this effect is absent with at
most. It also appears that Penka’s account inherits from Nouwen’s the problems we have
identified in section 3 regarding NPI licensing and the corresponding inferences.
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Bernhard Schwarz, Brian Buccola, and Michael Hamilton
This fact suggests a possible way to account for the bottom-of-the-scale
effect. Suppose that Nouwen’s lexical entry for up to in (9) above is strengthened so as to yield the implication that the relevant set of degrees contains
more than one element. Let us assume for concreteness that this implication is part of the asserted content. We then arrive at the revised lexical
entry in (34), where range(f), the range requirement, expresses that (the set
characterized by) f has more than one element.13,14
(34)
‚up toƒ = λdd .λfdt .max(f) = d & range(f)
Under our revised entry for up to, the truth conditions of (23), repeated
below, are the conjunction of the equality in (32) and the range requirement
in (35); likewise, the truth conditions of (28), also repeated below, are the
conjunction of the equality in (33) and the range requirement in (36).
(23)
[[up to] one] λd[♦ [[d many] people died in the crash]]
(35)
range({d : ♦∃!X[ |X| = d & ‚peopleƒ(X) = ‚died in the crashƒ(X) = T ]})
(28)
[[up to] half a dozen] λd[♦ [he bought [d many] eggs]]
(36)
range({d : ♦∃!X[ |X| = d & ‚eggsƒ(X) = ‚he boughtƒ(X) = T ]})
The LFs in (23) and (28) are now contradictory: In their respective truth
conditions, one conjunct implies that the relevant set of degrees is a singleton set, while the other implies that it is not. Under the assumption that
contradictory truth conditions can be a source of unacceptability, a view recently defended in Gajewski 2003, the range requirement therefore correctly
excludes (22b) and (27b) and more generally derives the bottom-of-the-scale
effect.
More accurately, so far we have only shown that the particular LFs in (23)
and (28) are contradictory. A complete explanation of the unacceptability
of the corresponding examples will have to exclude other potential LFs for
these examples as well. In particular, we will have to exclude LFs that, unlike
(23) and (28), feature no possibility modal in the scope of the degree phrase.
13 It should be clear that under this revised entry, up to is still not downward entailing (in
fact, it is still non-monotone), so the relevant observations in section 3 are accounted for as
before.
14 In Nouwen 2008, up to is a universal quantifier over degrees in an interval. Under this
view, the range requirement could be taken to follow from the general requirement that
the restrictor of a universal quantifier have more than one element (see, for example,
Schwarzschild 2002 for discussion).
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Two types of class B numeral modifiers
These are indeed excluded as well, as we will show in the next section. Just
like in Nouwen’s account, under the amendment we have proposed, LFs that
feature no possibility modal in the scope of up to will not give rise to attested
interpretations.
5.2
The range requirement and blocking
Introducing the range requirement forces us to abandon Nouwen’s (2010)
rationale for the obligatory presence of a silent possibility operator. We can
illustrate this with (21b), repeated below.
(21b)
Up to ten people died in the crash.
Recall that Nouwen’s account rests on the assumption that an LF can be
blocked by the availability of a semantically equivalent LF that counts as less
marked. For Nouwen, the LF for (21b) given in (37) is true just in case exactly
ten people died in the crash.
(37)
[[up to] ten] λd[[d many] people died in the crash]
Nouwen’s account ensures that the bare numeral example in (38a), or rather
its LF in (38b), is likewise true just in case exactly ten people died in the crash
and hence is equivalent to (37). Moreover, Nouwen assumes that (37) is more
marked than (38b), and so the latter blocks the former.
(38)
a.
b.
Ten people died in the crash.
[ten many] people died in the crash
It should be clear that under the revised lexical entry in (34), the LF in
(37) is no longer equivalent to (38b). The addition of the range requirement
renders (37) strictly stronger than (38b). Therefore, assuming that semantic
equivalence is a precondition on blocking, an alternative explanation must
be sought for the unavailability of (37).
Fortunately, such an alternative explanation is readily available. The LF in
(37) is, in fact, not merely stronger than (38b), but actually has contradictory
truth conditions. To see why, consider the set given by the degree predicate
in (37), shown in (39).
(39)
{d : ∃!X[ |X| = d & ‚peopleƒ(X) = ‚died in the crashƒ(X) = T ]}
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Bernhard Schwarz, Brian Buccola, and Michael Hamilton
This set can never have more than one element: If no one died in the crash,
it will be empty, and it will contain only the exact number of fatalities if
someone did; so the range requirement in (40) is necessarily false.
(40)
range({d : ∃!X[ |X| = d & ‚peopleƒ(X) = ‚died in the crashƒ(X) = T ]})
Therefore, under the assumption (already employed above) that contradictory
truth conditions can keep an LF from being attested, the effect of blocking in
Nouwen’s account is replicated.
Moreover, the addition of a possibility operator can be shown to result in
an LF whose truth conditions correspond to an observable interpretation. The
reason is that a possibility operator renders an otherwise contradictory range
requirement satisfiable. This will become clear in the next subsection, which
examines the effect of the range requirement on attested interpretations of
acceptable up to sentences.15
5.3
The range requirement and ignorance implications
In this section we have so far demonstrated the utility of the range requirement in preventing overgeneration by excluding problematic LFs containing bottom-of-the-scale numerals or lacking possibility modals. The range
requirement was seen to exclude the LFs in question by rendering them
contradictory. But surely the range requirement will have an effect on the
observable interpretations derived for acceptable up to sentences, and this
effect remains to be identified.
Let us therefore return to sentence (21b), repeated below. The sentence
has the intended LF in (41). The truth conditions we assign to this LF are
given by the conjunction of (42a) and (42b).
(21b)
Up to ten people died in the crash.
(41)
[[up to] ten] λd[♦ [[d many] people died in the crash]]
(42)
a.
b.
max({d : ♦∃!X[ |X| = d & ‚peopleƒ(X) &
‚died in the crashƒ(X) = T ]}) = 10
range({d : ♦∃!X[ |X| = d & ‚peopleƒ(X) &
‚died in the crashƒ(X) = T ]})
15 This account of the modal interpretation of class B modifiers is very similar to MenéndezBenito’s (2010) analysis of so-called universal free choice items.
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Two types of class B numeral modifiers
The conjunct in (42a), according to which 10 is the maximal number of
fatalities in the crash consistent with the speaker’s evidence, is of course an
entailment of the LF in (41) under Nouwen’s (2010) original account as well.
We accept Nouwen’s assumption that this is indeed an attested entailment of
sentence (21b).
Turning to (42b), let us attend to the set serving as the argument of range.
This set is isolated in (25), repeated below.
(25)
{d : ♦∃!X[ |X| = d & ‚peopleƒ(X) = ‚died in the crashƒ(X) = T ]}
This set contains all non-zero degrees that could be the exact number of
fatalities given the speaker’s evidence. To be sure, since it is possible for this
set to contain more than one element, the range requirement in (42b) can be
true. This supports our conclusion at the end of the previous subsection that
the addition of a possibility operator can render an otherwise contradictory
LF contingent.
Moreover, saying that the set in (25) has more than one element amounts
to saying that the set of worlds consistent with the speaker’s evidence varies
in terms of the exact number of people who died in the crash. Thus, (42b)
amounts to the statement that the speaker considers it possible that there
were fatalities in the crash, but does not have sufficient evidence to identify
the exact number. In other words, (42b) entails speaker ignorance regarding
the exact number of fatalities.
Sentence (21b) is indeed judged to imply such speaker ignorance. Of
course, this is an instance of Nouwen’s observation, reviewed in section
1, that all so-called class B modifiers introduce such implications. Recall
that Nouwen attributes the oddness of example (2b), repeated below, to the
implausibility of speaker ignorance regarding the number of sides a hexagon
has.
(2b)
#A hexagon has up to 10 sides.
Sentence (21b) illustrates the same effect, though perhaps in a less dramatic way, given that there is nothing implausible about the implied speaker
ignorance regarding fatalities in the crash.
We therefore conclude that in addition to accounting for the bottom-ofthe-scale effect, the range requirement has the welcome effect of deriving
attested ignorance implications contributed by up to, which under Nouwen’s
original proposal remained unaccounted for.
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Bernhard Schwarz, Brian Buccola, and Michael Hamilton
This is a pleasing result; however, in outlining Nouwen’s analysis of upperbound class B modifiers in section 2, we have permitted ourselves a slight
departure from his actual account. While inconsequential for Nouwen’s analysis of numeral modifiers, this departure raises questions for our modification
of it. This is the topic of the next section.
6
Weak many
We reported that Nouwen (2010) posits a counting quantifier, many, that
combines with numerals to form degree phrases. Actually, however, Nouwen
assumes that there are two such counting quantifiers (see footnote 2). In
addition to the many in (5), repeated below, Nouwen also posits a second
many, given in (43), which we write as manyw .
(5)
‚manyƒ = λdd .λfet .λget .∃!X[ |X| = d & f(X) = g(X) = T ]
(43)
‚manyw ƒ = λdd .λfet .λget .∃X[ |X| = d & f(X) = g(X) = T ]
The subscript w is meant to signal that the manyw in (43) derives truth
conditions that are weaker than those derived by the strong many in (5).
Consider again example (3) and the LF in (4), repeated below. We have seen
that (4) is true just in case exactly three students attended. In contrast,
(43) ensures that (44) is true just as long as no fewer than three students
attended.
(3)
Three students attended.
(4)
[[three many] students] attended
(44)
[[three manyw ] students] attended
Postulating manyw in (43) is motivated by the well-known observation that
sentences with bare numerals indeed allow for such weak interpretations (see
Geurts 2006 and the references provided therein). In this section, we describe
the consequences of introducing manyw for Nouwen’s original account as
well as for our own modification of it. We show that it is innocuous for the
former, but not for the latter. Specifically, we show that while our account
of the bottom-of-the-scale effect is preserved, our derivation of ignorance
implications is not. Note that we will not solve this problem. The purpose of
this section is merely to identify the price to pay for the kind of modification
to Nouwen’s account that we have proposed.
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Two types of class B numeral modifiers
6.1 Nouwen’s account revisited
Nouwen (2010) carefully demonstrates that when the two counting quantifiers
combine with upper-bound class B modifiers, the semantic difference between the strong version in (5) and the weak version in (43) is systematically
neutralized. It will be useful to review the reason for this neutralization by
attending again to (21b), repeated below. Consider first the LF with many in
(37), repeated below, and its counterpart with manyw in (45).
(21b)
Up to ten people died in the crash.
(37)
[[up to] ten] λd[[d many] people died in the crash]
(45)
[[up to] ten] λd[[d manyw ] people died in the crash]
Consider the sets denoted by the derived degree predicates in (37) and (45),
shown in (39) (repeated from earlier) and (46).
(39)
{d : ∃!X[ |X| = d & ‚peopleƒ(X) = ‚died in the crashƒ(X) = T ]}
(46)
{d : ∃X[ |X| = d & ‚peopleƒ(X) = ‚died in the crashƒ(X) = T ]}
Suppose that there have been fatalities in the crash and let f be their exact
number. We have seen that (39) is {f}. In contrast, (46) is the interval [1, f],
which apart from f includes all lower cardinalities down to the bottom of
the scale, 1. This implies that the two sets in (39) and (46) necessarily have
the same maximum. Accordingly, the two statements in (47) and (48) are
equivalent, for any cardinality n, and therefore under Nouwen’s semantics
for up to, the LFs in (37) and (45) have identical truth conditions.
(47)
max({d : ∃!X[ |X| = d & ‚peopleƒ(X) = ‚died in the crashƒ(X) = T ]}) = n
(48)
max({d : ∃X[ |X| = d & ‚peopleƒ(X) = ‚died in the crashƒ(X) = T ]}) = n
The same is true for the LFs in (41) (repeated from earlier) and (49), where
the silent possibility modal has been added to the scope of up to.
(41)
[[up to] ten] λd[♦ [[d many] people died in the crash]]
(49)
[[up to] ten] λd[♦ [[d manyw ] people died in the crash]]
Now the derived degree predicates denote the sets in (25) (repeated from
earlier) and (50).
(25)
{d : ♦∃!X[ |X| = d & ‚peopleƒ(X) = ‚died in the crashƒ(X) = T ]}
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Bernhard Schwarz, Brian Buccola, and Michael Hamilton
(50)
{d : ♦∃X[ |X| = d & ‚peopleƒ(X) = ‚died in the crashƒ(X) = T ]}
Suppose that according to the speaker’s evidence, someone may have
died in the crash and let p be the largest number of fatalities consistent with
that evidence.16 Then (25) is a set whose maximal element is p and (50) is the
set [1, p]. This entails once again that the sets in (25) and (50) have the same
maximum element. For any cardinality n, the statements in (51) and (52) are
equivalent, and so (41) and (49) have identical truth conditions.
(51)
max({d : ♦∃!X[ |X| = d & ‚peopleƒ(X) & ‚died in the crashƒ(X) = T ]})
=n
(52)
max({d : ♦∃X[ |X| = d & ‚peopleƒ(X) & ‚died in the crashƒ(X) = T ]})
=n
The addition of manyw , then, is innocuous for Nouwen’s original account
of upper-bound class B modifiers. Unfortunately, this welcome feature is not
shared by the revised version we have presented above. While our account of
the bottom-of-the-scale effect is preserved under the addition of manyw , our
rationale for why up to must be accompanied by a possibility operator must
be revised, and our proposed derivation of ignorance implications is lost. We
dedicate the following three subsections to substantiating these points.
To facilitate the presentation, though, it will be useful to first highlight
another pair of equivalencies. Recall that the sets in (46) and (50), the sets
arising from LFs with manyw , necessarily denote intervals of the form [1, n],
sets consisting of a maximum n and all smaller cardinalities down to the
bottom of the scale, 1. This means that the membership of these sets is fully
determined by their maximal elements: For any cardinality n, the maximality
statements in (48) and (52) are equivalent to the equalities in (53) and (54).
(53)
{d : ∃X[ |X| = d & ‚peopleƒ(X) = ‚died in the crashƒ(X) = T ]} = [1, n]
(54)
{d : ♦∃X[ |X| = d & ‚peopleƒ(X) = ‚died in the crashƒ(X) = T ]} = [1, n]
Having a name for these two equivalencies will simplify the exposition below,
so we will collectively refer to them as the range law.
16 The sets in (25) and (50), even if non-empty, are not guaranteed to have maximal elements.
However, we can safely focus on the case where they do, as this is a presupposition triggered
by up to in Nouwen’s account.
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Two types of class B numeral modifiers
6.2 Bottom-of-the-scale effect revisited
Above we proposed that (22b), repeated below, is excluded because it only
has a contradictory interpretation. To maintain this account, we now have to
consider the LF with manyw in (55). We want to show that this LF shares the
contradictory truth conditions of its counterpart with (strong) many (see (23)
in section 5.1).
(22b) #Up to one person died in the crash.
(55)
[[up to] one] λd[♦ [[d manyw ] people died in the crash]]
Given the semantics of manyw , the truth conditions we assign to (55) are given
by the conjunction of (56a) and (56b).
(56)
a.
b.
max({d : ♦∃X[ |X| = d & ‚peopleƒ(X) =
‚died in the crashƒ(X) = T ]}) = 1
range({d : ♦∃X[ |X| = d & ‚peopleƒ(X) =
‚died in the crashƒ(X) = T ]})
Just as we saw in section 5.1 with (strong) many, a set of degrees whose
maximum is the bottom of the scale is a singleton set containing that degree.
It follows that (56a) is equivalent to (57). (This is also evident from the range
law since [1, 1] = {1}.)
(57)
{d : ♦∃X[ |X| = d & ‚peopleƒ(X) ‚died in the crashƒ(X) = T ]} = {1}
Therefore the conjunction of (56a) and (56b) is inconsistent, hence the LF in
(55) is excluded in the same way as its variant with strong many in (23).
It should moreover be clear that omitting the possibility operator from
(55) does not change the relevant semantic relations: The maximum of the
relevant set is still the bottom of the scale (and the range law will apply as
before), and so a contradiction will again be derived. The addition of manyw ,
then, does not save bottom-of-the-scale numerals from yielding contradictory
truth conditions, and so we conclude that our account of the bottom-of-thescale effect still applies correctly.
6.3 Blocking revisited
Nouwen (2010) proposes that unless they contain a possibility modal (in
the immediate scope of the degree phrase), LFs with upper-bound class B
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Bernhard Schwarz, Brian Buccola, and Michael Hamilton
modifiers are blocked by less-marked LFs with bare numerals. In our revised
account, only positing strong many, we proposed that instead of being
blocked by markedness, the relevant LFs are excluded by virtue of being
contradictory. Having added manyw , we now have to revisit this account.
We will focus here on numerals above the bottom of the scale, since we just
discussed how bottom-of-the-scale numerals give rise to contradictions.
Consider again example (21b), repeated below. We need to examine the
predicted interpretation of the LF in (45), repeated below, which features
manyw but no possibility modal.
(21b)
Up to ten people died in the crash.
(45)
[[up to] ten] λd[[d manyw ] people died in the crash]
Under our semantics for up to, the truth conditions of (45) are the conjunction
of (58a) and (58b).
(58)
a.
b.
max({d : ∃X[ |X| = d & ‚peopleƒ(X) = ‚died in the crashƒ(X) = T ]})
= 10
range({d : ∃X[ |X| = d & ‚peopleƒ(X) = ‚died in the crashƒ(X) = T ]})
Note that according to the range law, (58a) is equivalent to (59). Since in this
case the numeral is above the bottom of the scale, the set of cardinalities it
determines according to the range law is necessarily a range.
(59)
{d : ∃X[ |X| = d & ‚peopleƒ(X) = ‚died in the crashƒ(X) = T ]} = [1, 10]
Accordingly, the maximality statement in (58a) entails the range requirement
in (58b). It follows that the conjunct in (58b) is effectively redundant: The
truth conditions of (45) are given by (58a) alone.
So our revised semantics for up to assigns to (45) the very same truth
conditions as Nouwen’s original semantics: The LF is true just in case exactly
ten people died in the crash. Since these truth conditions are contingent,
our rationale for why up to requires a possibility operator in its scope is not
preserved under the addition of manyw .
As things stand, then, we are in danger of predicting an unattested
interpretation of (21b), according to which exactly ten people died in the
crash. Evidently, we must abandon the idea that the presence of a possibility
operator is always motivated by the need to prevent contradiction.
Fortunately, though, given that we assign to (45) the very same truth
conditions as Nouwen, a natural modification suggests itself. To force the
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Two types of class B numeral modifiers
presence of a possibility operator in an LF with manyw , we can adopt Nouwen’s
notion of blocking by markedness, which after all is not inconsistent with
anything we have proposed. So, following Nouwen, we could assume that
(45) is excluded by virtue of being more marked than the equivalent LF in
(38b), repeated from earlier.
(38b)
[[ten many] people died in the crash]
In the resulting account, then, there would be two different reasons why
up to requires the presence of a possibility operator (with numerals above
the bottom of the scale): In LFs with strong many, the possibility operator
renders an otherwise contradictory LF contingent; in LFs with manyw , the
possibility operator prevents the LF from being blocked by a less-marked
competitor.
6.4 Ignorance implications revisited
We have proposed to credit ignorance implications to the range requirement
introduced by the lexical semantics of up to. Once again, we need to go back
and examine the effect of the addition of manyw . Let us return to the LF in
(49), repeated below.
(49)
[[up to] ten] λd[♦ [[d manyw ] people died in the crash]]
Under our account, the truth conditions of (49) are given by the conjunction of (60a) and (60b). According to the range law, we moreover have the
equivalence of (60a) and (61).
(60)
a.
b.
(61)
max({d : ♦∃X[ |X| = d & ‚peopleƒ(X) =
‚died in the crashƒ(X) = T ]}) = 10
range({d : ♦∃X[ |X| = d & ‚peopleƒ(X) =
‚died in the crashƒ(X) = T ]})
{d : ♦∃X[ |X| = d & ‚peopleƒ(X) = ‚died in the crashƒ(X) = T ]} = [1, 10]
This means that much like in the case studied in the previous subsection, the
range requirement ends up being redundant, as it is entailed by (60a).17 So in
17 Despite initial appearance, the statement in (61) does not imply speaker ignorance. Specifically, it is consistent with the speaker being certain that exactly ten people died in the crash.
So (61) does not entail that the set of worlds consistent with the speaker’s evidence varies
in terms of exact number of fatalities. In contrast, equating the set in (25) with a range like
1:21
Bernhard Schwarz, Brian Buccola, and Michael Hamilton
this case, too, our semantics for up to derives the very same truth conditions
as Nouwen’s.
This is certainly not a welcome result. The range requirement being
redundant, our account of ignorance implications is lost. We showed that
the range requirement derives ignorance implications for LFs with strong
many. However, if we assume that manyw is freely available as well, we admit
LFs where the range requirement does not actually strengthen the truth
conditions in any way, and so we no longer predict that up to is perceived to
always introduce these implications.
So if one admits manyw , an alternative derivation of ignorance implications needs to be sought. But insisting on deriving ignorance implications
through the range requirement, by excluding manyw , has its price as well. As
reported above, manyw is motivated by weak readings of bare numerals, so
an alternative account of those would be needed.18
7
Summary and conclusion
What we consider the main result of this commentary is that so-called upperbound class B numeral modifiers come in two types. This is shown, first,
by NPI licensing facts and intuitions on corresponding inferences. The
class B numeral modifier at most licenses NPIs and is downward entailing,
according to intuitions on inferences. In contrast, the class B modifier up
to has neither of these properties. Second, while at most can combine with
bottom-of-the-scale numerals, up to cannot. These basic semantic contrasts
suffice to establish that at most and up to cannot share the same semantic
interpretation.
More specifically, NPI licensing data and the corresponding inferences
demonstrate that Nouwen’s (2010) semantics for upper-bound class B modifiers is not correct for at most. In attempting to apply it to up to, we identified
[1, 10] does imply speaker ignorance.
18 The literature provides possible starting points for either direction of analysis. One might
hope to derive ignorance implications as conversational implicatures, perhaps along the
lines of Büring 2008, which provides an analysis of at least (see also Cummins & Katsos 2010
and Schwarz & Shimoyama 2011). Conversely, one might follow Kratzer 2003 in employing
situation semantics to derive apparent weak readings from strong lexical meanings, such as
the strong many we have been discussing. A third possibility is that manyw is an available
lexical item with a suitably limited distribution: Specifically, it cannot combine with a trace
of a numeral modified by up to; unfortunately, however, we don’t see what principle of
grammar such a restriction might follow from.
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Two types of class B numeral modifiers
two shortcomings. First, the analysis fails to derive the bottom-of-the-scale
effect; second, it surprisingly fails to derive the very ignorance implications
that Nouwen identified as a defining feature of class B modifiers.
We presented an amendment of Nouwen’s account that adds the range
requirement to the truth conditional content of up to. This amendment
captures the bottom-of-the-scale effect. Under the additional assumption
that the counting quantifier many is always strong, but not without this
assumption, we also derive ignorance implications. So the extent to which
our proposal improves upon Nouwen’s original account depends on whether
a weak counting quantifier is assumed to be available.
Apart from the issue raised by manyw , one additional concern about our
proposed derivation of ignorance implications with up to is that it might
be insufficiently general, as it is unclear how it would carry over to at most.
However, in view of the semantic differences between up to and at most
that we have identified, it is not clear to us that the ignorance implications
associated with the two expressions should even have the same source. Thus,
our discussion raises the possibility that there are two types of ignorance implications, and hence that this common feature of class B numeral modifiers
is not necessarily indicative of a shared analysis.
To be sure, there may well be subclasses of class B modifiers which do
form semantic and pragmatic natural classes. If morphology is a guide, we
expect at least to pattern with at most (see Penka 2010). But there are other
numeral modifiers characterized as class B by Nouwen, such as minimally
and maximally, whose place in the typology remains to be determined.
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Bernhard Schwarz
Department of Linguistics
1085 Dr. Penfield
Montréal, Québec
H3A 1A7
Canada
[email protected]
Brian Buccola
Department of Linguistics
1085 Dr. Penfield
Montréal, Québec
H3A 1A7
Canada
[email protected]
Michael Hamilton
Department of Linguistics
1085 Dr. Penfield
Montréal, Québec
H3A 1A7
Canada
[email protected]
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